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The Howard University
Department of Physics and Astronomy
Master of Science Comprehensive and
Doctor of Philosophy Qualifying Exam
Written exam: Classical Physics
August 27, 2013
Attempt to solve four problems, at least one from each group. Circle
the numbers below to indicate the problems you chose to solve:
123
Group A
45
Group B
67
Group C
1. Write in your chosen code-letter here:
2. Write your code-letter and a page number (in sequential order)
on the top right-hand corner of each submitted answer sheet.
3. Write only on one side of your answer sheets.
4. Start each problem on a new answer sheet.
5. Stack your answer sheets by problem and page number
and then staple them (at the top left-hand corner ),
with this cover sheet on the top.
1
Howard University Physics MS Comprehensive/Ph.D. Qualifying Exam: Classical Physics
Problem 1.
Calculate the gravitational potential and gravitational force field on the axis of:
[6 pt]
a. A thin uniform disc of mass M and radius R.
[6 pt]
b. A thin uniform annulus of mass M having inner and outer radii R1 and R2 , respectively.
[6 pt]
c. Show that in the limit z R, the potential in part a reduces to −GM/z.
[7 pt]
d. Show that in the limit z R, the force field in part a reduces to a constant −2GM ẑ/R2
(for positive z).
Problem 2.
A uniform flexible chain of length l and mass M is initially suspended
with its two ends close together and at the same elevation, and then one
end is released. Consider a one dimensional approximation to this twodimensional problem, in which the chain is represented by two vertical
segments connected by a horizontal cross piece which is sufficiently short
that its contribution to the kinetic and potential energies may be neglected;
see figure to the right.
[6 pt]
a. Find the Lagrangian of the chain and obtain the Lagrange equations.
[6 pt]
b. Prove that the energy is conserved.
[7 pt]
c. Using conservation of energy, find the speed of the falling end
x(t)
as a function of its position, x.
[6 pt]
d. Find the tension force at the fixed end as a function of x.
Problem 3.
A thin uniform rod AB of mass M and length ` is placed with the
end A against the base of a wall and inclined at an angle θ0 to
the vertical; see figure to the right. A point mass m is attached
to the end B and the rod is released from rest to fall under the
influence of gravity.
[6 pt]
B
M
a. Find the moment of inertia of the rod with the point mass
about the axis of rotation.
[7 pt]
b. Find the Hamiltonian of the system.
[6 pt]
c. Find the Hamilton equations for the system.
[6 pt]
d. Find the time it takes the rod to reach the ground.
(The result can be expressed as quadratures.)
θ0
A
`
m
2
Howard University Physics MS Comprehensive/Ph.D. Qualifying Exam: Classical Physics
Problem 4.
An L × L square parallel-plate capacitor and separation d is
charged to a potential V and disconnected from the battery, then
vertically lowered into a large reservoir of dielectric liquid of relative dielectric constant , density ρ, and negligible surface tension,
until the liquid fills half the space between the plates; see figure
to the right.
[6 pt] a. Calculate the capacitance of the so-constructed capacitor.
d
L
h
[6 pt]
b. Calculate the electric field between the capacitor plates.
[6 pt]
c. Calculate the surface charge density on the plates.
[7 pt]
d. Calculate the height (vs. outside level) to which the liquid has risen between the plates above
ρ, the outside surface level.
Problem 5.
A perfectly conducting sphere of radius R is uniformly charged over its surface to a total charge Q,
and is rotating about its axis with a constant angular velocity ω.
[6 pt]
a. Compute the electric current distribution ~ (~
r ).
[6 pt]
~ A
~ = 0, determine the differential equations and boundary conditions that the
b. Assuming ∇·
~ r ) must satisfy.
vector potential A(~
[6 pt]
~ r ) inside and outside the sphere.
c. Using the cylindrical symmetry of the system, compute A(~
[7 pt]
~ = ∇×
~ A
~ inside and outside the rotating charged sphere.
d. Compute B
~ r ) = êφ Aφ (r, θ), so (∇
~ 2 A)
~ φ = [∇
~2−
[With cylindrical symmetry, choosing A(~
1
]Aφ
r2 sin2 θ
may be useful.]
Howard University Physics MS Comprehensive/Ph.D. Qualifying Exam: Classical Physics
3
Problem 6.
A system is composed of N identical subsystems that have energy 0 or . The total energy of the
system is equal to E0 .
[8 pt]
a. Given the thermodynamic limit, in identically equilibrium, find the temperature of the system
in terms of the quantities mentioned above.
[9 pt]
b. Find the entropy of a separate, second system, an ideal gas with NG particles in a volume V
at a temperature TG .
[8 pt]
c. Now, bring both systems into thermal contact. The composite system can be considered
isolated. Compute their equilibrium temperature in terms of the quantities given previously.
Problem 7.
Fluctuation Relations: For the Grand Canonical ensemble where κT is the isothermal compressibility
and kB is the Boltzman constant,
D
2 E
[12 pt] a. Prove:
N − hN i
= kB T n2 κT .
D
2 E
[13 pt] b. Prove:
H − µN − hH − µN i
= kB T 2 CV,µ .
The Howard University
Department of Physics and Astronomy
Master of Science Comprehensive and
Doctor of Philosophy Qualifying Exam
Written exam: Modern Physics
August 29, 2013
Attempt to solve four problems, at least one from each group. Circle
the numbers below to indicate the problems you chose to solve:
12
Group A
345
Group B
678
Group C
1. Write in your chosen code-letter here:
2. Write your code-letter and a page number (in sequential order)
on the top right-hand corner of each submitted answer sheet.
3. Write only on one side of your answer sheets.
4. Start each problem on a new answer sheet.
5. Stack your answer sheets by problem and page number
and then staple them (at the top left-hand corner ),
with this cover sheet on the top.
Howard University Physics MS Comprehensive/Ph.D. Qualifying Exam: Modern Physics
1
Problem 1.
An electron of total energy 1.40 MeV collides with another electron, initially at rest in the lab frame.
[6 pt]
a. Compute the total energy and momentum of the system in the lab frame.
[6 pt]
b. Compute the velocity of the center of mass (CM) in the lab frame.
[6 pt]
c. Compute the total energy of the pair of electrons in the CM frame.
[7 pt]
d. If the target electron scatters 45◦ from the direction of the incoming electron, compute the
direction and magnitude of the momentum of the incoming electron.
The rest energy of the electron is 0.511 MeV.
Problem 2.
Consider the lowest-order relativistic corrections to the linear harmonic oscillator, the non-relativistic
1
P 2 + 12 M ω 2 x2 .
Hamiltonian of which is H 0 = 2M
[6 pt]
a. By expanding the relativistic kinetic energy, obtain the first order relativistic correction to the
energy and use this to define the relativistic perturbation operator H r .
[6 pt]
b. Compute the correction to all energy levels to lowest nonzero order in perturbation theory.
[6 pt]
c. Prove the relation 2M c2 hn|H r |ni = 2En(0) hn|V |ni − hn|V 2 |ni − (En(0) ).
[7 pt]
d. Compute, independently, the correction to all energy levels to lowest nonzero order in perturbation theory from the anharmonic perturbation H 0 = λx4 , and verify that this result agrees
with your results in parts b and c.
Howard University Physics MS Comprehensive/Ph.D. Qualifying Exam: Modern Physics
2
Problem 3.
Two spin-1/2 electrons are constrained to the 1-dimensional interval x ∈ [0, L] and arranged to have
parallel spins, but otherwise move freely.
[6 pt]
a. Determine the properly normalized 1-electron wave-functions ψi (x) for both electrons, i = 1, 2.
[6 pt]
b. Using ψi (x), write down the most general wave-function for this 2-electron system.
[6 pt]
c. Compute the probability that both electrons are in the same half of the interval.
[7 pt]
d. Compute the probability that the two electrons are in different halves of the interval.
Problem 4.
A particle of mass M is moving in one dimension subject to the potential V (x) = +∞ for x < 0
and V (x) = −V0 a δ(x−a) for x > 0 and with V0 , a > 0.
[8 pt]
a. Determine all boundary conditions on the wave-function φ(x) for this particle.
[8 pt]
b. Determine the general solution of the Schrödingier equation, and all available conditions on
the integration constants and the energy.
[9 pt]
c. Determine for which values of V0 , a can there exist a bound state, and compute the critical
value of V0 (with a fixed) for which the ground state energy vanishes.
Problem 5.
A spinless particle of mass M and charge q is restricted to the (x, y)-plane, where it experiences
~ = B0 êz .
the harmonic potential 12 M ω 2 (x2 +y 2 ) and the constant perpendicular magnetic field B
[6 pt]
a. Determine the Hamiltonian operator governing the dynamics of this particle.
[6 pt]
b. Prove that the Schrödinger equation may be solved in the polar form, ψ = N f (ρ)eimφ , and
determine the differential equation for f (ρ).
[6 pt]
c. Solve this equation exactly for the wave-functions and the energies.
[7 pt]
d. Considering now B0 as small, compute the lowest order perturbative correction to the energies
and compare with the exact result.
[Toggling between Cartesian and polar coordinates may prove useful.]
Howard University Physics MS Comprehensive/Ph.D. Qualifying Exam: Modern Physics
3
Problem 6.
Two spin-1/2 particles are separated by a distance ~a = a êz and interact only through the magnetic
dipole Hamiltonian H = a13 µ
~ 1 ·~µ2 − a35 (~µ1 ·~a)(~µ2 ·~a), where µ
~ i is the magnetic moment of the ith
~i .
particle, assumed to be proportional to its spin, S
[6 pt]
~ i.
a. Specify the Hamiltonian operator H in terms of the spin-operators S
[6 pt]
~ := S~ 1 + S~ 2 , specify H in terms of the spin-operators S 2 and S z .
b. Defining S
[6 pt]
c. Compute the energies for all the states of this system.
[7 pt]
~ = B0 êz is turned on, determine the new Hamiltonian operator
d. If an external magnetic field B
H tot = H + HB~ , and compute the shifted energies in this system.
Problem 7.
A plane-wave of energy E is incident upon a axially symmetric potential V = 0 for ρ > a, V = V0 > 0
for ρ 6 a, and scatters off of it.
[5 pt]
a. Write down the Schrödinger equation, separate variables, and
[5 pt]
b. Determine the general solutions for E < V0 and E > V0 , and all boundary conditions.
[5 pt]
c. Derive the exact condition specifying the phase-shift δ0 in the scattered S-wave.
[5 pt]
d. Determine the behavior of δ0 when V0 → ∞.
[5 pt]
e. Determine δ0 when E → 0, and compute the corresponding total cross-section.
[Recall that for central potentials, the substitution R(r) =
u(r)
r
was useful.]
Problem 8.
There is a model of the thermal behavior of crystalline solids, according to which each of the N
atoms of the solid behaves like three independent harmonic oscillators. The 3N harmonic oscillators
(which are on distinguishable sites) all have the same frequency, ω0 .
[8 pt]
a. Write an expression for the possible energy levels n of the nth harmonic oscillator.
[8 pt]
b. Calculate an expression for the Helmholtz free energy, F (N, T ), of the solid.
[8 pt]
c. Find the heat capacity, C(T ), as a function of the temperature. Give the general formula and,
to leading order, show explicitly how to obtain the low-temperature and high-temperature
limits. Sketch a plot of C versus T .